Lemma 39.7.6.label Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P, Q \in \text{Proj}(A)$ with $P \sim Q$, then:
- (1)
$Z(P) = Z(Q)$.
- (2)
For any central projection $R \in \text{Proj}(A)$, $PR \sim QR$.
Proof. Let $V \in A$ with $P = V^{*}V$ and $Q = VV^{*}$, then $V$ is a partial isometry with initial space $P(H)$ and final space $Q(H)$.
(1): Since $Z(P) \ge P$ and $Z(P) \in Z(A)$,
\[Z(P)Q = Z(P)VV^{*} = VZ(P)V^{*} = VV^{*} = Q\]
and $Z(P) \ge Q$, and $Z(P) \ge Z(Q)$. By symmetry, $Z(Q) \ge Z(P)$, so $Z(P) = Z(Q)$.
(2): Let $R$ be a central projection, then
\[PR = V^{*}VR = V^{*}RV = (VR)^{*}(VR) \sim (VR)(VR)^{*} = VRV^{*} = VV^{*}R = QR\]
$\square$
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